We’re going to get a little clearer on the notions of ‘form’ and ‘structure’ from last time.
Associated Reading
forall x, chapter 4, “First Steps to Symbolization”.
Definition
An argument is valid when
The truth of the premises guarantees the truth of the conclusion.
Equivalently, when it is impossible for the premises to be true and the conclusion not true.
Formal Validity
An argument is formally valid if
It is valid; and
Any argument of the same form is valid.
In those cases, we infer that it is valid because of its form.
Three Big Questions About Form
Which words get treated as part of the form, and which get replaced by symbols when converting into formal notation?
Which words have the same meaning and should be replaced by the same symbol, and which should be treated differently?
How do we break the sentence into its constituent parts?
Part of the Form
I’m going to have a fairly pluralist, pragmatist view on this question.
This is not to say that pluralism or pragmatism is right - it’s just that it’s what I’ll adopt. (It is to say that I think it’s a reasonable approach.)
For some purposes the ‘formal words’ are very few, like ‘and’, ‘or’, ‘not’ etc.
For other purposes the set is more expansive.
Part of the Form
If you are studying the logic of causation, you might want to say that ‘caused’ is a formal word.
And that would mean any two arguments that have this structure are of the same form.
A caused B.
B caused C.
\(\therefore\)
A caused C.
Part of the Form
But for other purposes, ‘caused’ is just another binary relation between events.
So those arguments will have the same form as these ones.
A happened near B.
B happened near C.
\(\therefore\)
A happened near C.
Part of the Form
What I mean by having a ‘pluralist, pragmatist’ view about form is that there is no objectively correct answer about whether these are the same form or not.
For some purposes it will be helpful to treat them as having the same form.
For other purposes it will not be.
We’re going to start with a notion of form where lots and lots of things have the same form, so very few arguments are valid in virtue of form.
Three Big Questions About Form
Which words get treated as part of the form, and which get replaced by symbols when converting into formal notation?
Which words have the same meaning and should be replaced by the same symbol, and which should be treated differently?
How do we break the sentence into its constituent parts?
Same/Different
There are problems in both directions here.
Sometimes we want to treat words with different spelling as representing the same meaning - e.g., ‘kangaroo’ and ‘kangaroos’.
And sometimes we want to treat words with the same spelling as expressing different meanings.
Relating the First Two Questions
Some kinds of ambiguity are ‘resolved’ by treating the word as part of the formal system. Consider two possible interpretations of this sentence.
Nothing is better than world peace
“Nothing is better than world peace”
There is no thing such that it is better than world peace.
It is better that there is nothing than that there is world peace.
Ambiguity and Form
If you include words like ‘nothing’ as formal words, then the two readings of the sentence will have different formal representations.
And the logic called ‘predicate logic’ does exactly that; it treats ‘nothing’ as one of the formal words.
We’re not covering that in this course, though some of you have covered it elsewhere.
Some more expansive accounts of what should be a formal word are moved by the thought that treating more words as formal removes some possible ambiguities.
Three Big Questions About Form
Which words get treated as part of the form, and which get replaced by symbols when converting into formal notation?
Which words have the same meaning and should be replaced by the same symbol, and which should be treated differently?
How do we break the sentence into its constituent parts?
Three Formal Tools
Replacing words/sentences with letters.
Using parentheses.
Replacing ‘formal words’ with symbols.
Formal Tools
Overview
We’re going to look at how to translate English sentences into a form where we can use the tools of propositional logic on them.
Associated Reading
forall x, chapter 5, “Connectives”
Aside
The book goes into more detail than I’m going into here about the range of English sentences that can be translated (or at least symbolized) using the tools of propositional logic.
Sentence Letters
We use letters, usually starting with \(p\) and then continuing through the alphabet, for ‘atomic’ sentences.
forall x uses capital letters, others use lowercase. This is a matter of taste.
Symbols
“and” is represented by \(\wedge\)
“or” is represented by \(\vee\)
“not” is represented by \(\neg\)
“if” is represented by \(\rightarrow\)
“if and only if” is represented by \(\leftrightarrow\)
An Easy Example
It will rain on Monday or it will rain on Tuesday
Basic Sentences
\(M\) = It will rain on Monday
\(T\) = It will rain on Tuesday
Translation
\(M \vee T\)
Another Easy Example
It will rain on Monday and it will rain on Tuesday
Basic Sentences
\(M\) = It will rain on Monday
\(T\) = It will rain on Tuesday
Translation
\(M \wedge T\)
Only Slightly Harder Example
If it rains on Monday, it will rain on Tuesday
Basic Sentences
\(M\) = It will rain on Monday
\(T\) = It will rain on Tuesday
Translation
\(M \rightarrow T\)
And A Bit Harder Again
It will rain on Monday, but it won’t rain on Tuesday
Basic Sentences
\(M\) = It will rain on Monday
\(T\) = It will rain on Tuesday
Translation
\(M \wedge \neg T\)
Parentheses
As in math, these tell you what order to compute things in.
They let you avoid the ambiguity in the following sentence.
It will rain on Monday and it will rain on Tuesday or it will rain on Wednesday.
Two Translations
Basic Sentences
\(M\) = It will rain on Monday
\(T\) = It will rain on Tuesday
\(W\) = It will rain on Wednesday
Translations
\((M \wedge T) \vee W\)
\(M \wedge (T \vee W)\)
What’s the Difference?
\((M \wedge T) \vee W\)
\(M \wedge (T \vee W)\)
What is a situation where one of these is true but not the other?
If it only rains on Wednesday, then 1 is true but 2 is false.
Are there any others?
Recursion
Overview
We’re going to look at how and why we can iterate the translation procedures we’ve been investigating.
Associated Reading
forall x, chapter 6, “Sentences of TFL”
Recursion
The language of propositional logic has some fairly simple composition rules.
It says what the basic sentences are.
It has some rules saying that if some things are sentences, so are some other things.
The effect is that there are an infinity of possible sentences.
Natural Language Recursion
As speakers of a human language, you’re used to this kind of recursion. All of these are sentences.
It will rain.
Alex thinks that it will rain.
Kim thinks that Alex thinks that it will rain.
Alex thinks that Kim thinks that Alex thinks that it will rain.
And so on, to infinity, without adding any more words.
Recursive Rule
If \(S\)$ is a sentence, and \(N\)$ is a name, then \(N\)thinks that\(S\) is a sentence.
Note that the output of this rule can be the input to a new instance of it.
Formal Language Recursion
The letters \(P, Q, R...\) are sentences.
If \(S\) and \(T\)$ are sentences, then so are:
\(\neg S\)
\(S \vee T\)
\(S \wedge T\)
\(S \rightarrow T\)
\(S \leftrightarrow T\)
Multiple Steps
So these are all sentences. (Note that I’m playing fast and loose with parentheses here.)